Improper Signaling in the Complex Multicarrier MIMO BC-TIN

Let us first consider the power shaping spacesPk = `PUk, ∀kandPB = `PB, i.e., we study potential benefits of improper signaling.

7.4.1.1 Benefits of Improper Signaling

The following result is an extension of [16, Th. 1].

Proposition 7.4.1. In the complex MIMO BC-TIN with proper noise and a sum power constraint (3.4), proper per-user input signals do not always achieve the whole rate region. This statement holds with and without zero-forcing constraints and with and without (R)TS.

Proof. Consider the combined real representation (3.37) of the complex three-user MISO BC-TIN with noise covariance matricesC_ηˇk = ¹₂I₂, a sum power constraint withQ= 5, and

Improper Signaling—Achievability: Let the transmit covariance matrices in the dual uplink be ρ_improper,k≈1.0237∀kare achievable without zero-forcing, while (7.23) withµ= ¹₂ yields thatρZF,improper,k≈1.0107∀kis achievable with zero-forcing. These rates are of course still achievable if (R)TS is allowed.

Proper Signaling—Converse: The corresponding complex channels are row vectors h₁^H=h

7.4. Benefits of Reduced-Entropy Transmission 105

0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2

r₁ r2=r3

optimal proper without (R)TS optimal proper with TS achievable improper without (R)TS upper bound (capacity region)

Figure 7.2: Achievable rates in the complex three-user MISO BC-TIN from the proof of Proposition 7.4.1 (intersections of the rate regions with the planer2 =r3). The diamonds correspond to the values calculated in the proof of Proposition 7.4.1, and the gray diagonal line indicates the points where all users achieve the same rate.

and the proper noise of each userkis a scalar with varianceCη_k = 1. Consider a dual uplink transmission with three equal-length time slots, where a different pair of two users is served with equal powerCx_k = ^Q₂ in each time slot. Then, the overall rate of each user is ²₃ of the per-slot rate, which we can calculate using (7.14) withµ= 1. We obtainρ_proper,k≈0.92165∀k. By means of the method from Section 7.3.1.2, we can verify that this is indeed a Pareto-optimal TS solution. Thus,ρ_improper,k from above is not achievable with proper signaling, and it of course stays unachievable if we introduce zero-forcing constraints and/or consider only strategies without TS (where either RTS is used or (R)TS is avoided completely).

An illustration of the proof can be seen in Figure 7.2, where we have plotted intersections of various rate regions with the planer₂ = r₃. Note that the upper left end of the Pareto boundary thus corresponds to a transmission to two users while the lower right point describes a single-user transmission, where all schemes have equal performance. The capacity region, which is achievable with DPC (see Chapter 6), is included as an upper bound. To obtain (potentially suboptimal) achievable solutions with improper signaling, we have applied the rate balancing algorithm from [9] in the combined real representation using random initializations.3 Even without (R)TS, improper signaling brings a considerable improvement over the optimal proper strategy with TS nearly along the whole Pareto boundary. When combining improper signaling with TS, we could obtain a rate region that is at least as large as the convex hull of the gray curve.

7.4.1.2 Optimality of Proper Signaling in Special Cases

On the other hand, there are special cases of the MIMO BC-TIN, for which it can be shown that improper signaling can never bring a gain compared to proper signaling. One of them is given

3Algorithmic aspects are discussed in detail in Section 7.6.

in the following proposition.

Proposition 7.4.2. In the complex two-user single-carrier MISO BC-TIN with proper noise, the whole TS rate regionRTIN(Q)is achieved by proper per-user input signals.

Proof. Let%_1,1 ∈Rand%_1,2, %_2,2∈Cbe the elements ofRfrom the reduced QR decomposition QR =H = [hUL,1,hUL,2]∈C^m×2. As the noise covariance matrix is an identity in the dual uplink, we can apply the following filter without changing the achievable rates:

y⁰= transformationx₂⁰ = e^−jθ2x2does not change the transmit power, andx₂⁰ is improper if and only ifx2is improper. We may chooseθandθ2such thatH⁰ = [h⁰₁,h₂⁰]∈R^2×2. The corresponding composite real channel matricesH`<sub>1</sub><sup>0</sup> = `<(h₁⁰)andH`<sub>2</sub><sup>0</sup> = `<(h₂⁰)are then block-diagonal with two equal blocks due to (2.46).

Any possible composite real uplink transmit covariance matrix can be parametrized as C_xˇ⁰ withj6=k. The following facts can be verified analytically (e.g., with the help of a software for symbolic calculations). The determinantsdetX_k,k∈ {1,2}do not depend onϕ1norϕ2. The determinantdetX does not depend onϕ₁andϕ₂individually, but only on the difference

∆=ϕ₂−ϕ₁, and we have where we have chosenϕ₁ = 0without loss of generality. The intuition behind this result is that the impropriety of the intended signal and of the interference should point exactly in opposite directions (cf. Example 4.5.1 or a similar result for a one-sided interference channel in [38] and Theorem 8.3.1).

We can thus assume thatC_xˇ⁰

k, ∀kin (7.56) are diagonal matrices. Together with the special structure of the channel matricesH`_k⁰, we can interpret this as CN transmission in a real-valued two-carrier system with equal channel vectors on both carriers.

To optimize the CN transmission, we can apply the dual decomposition from Section 3.3.2.

Since the channel vectors are equal on both carriers, the inner problem (3.33) is exactly the same on both carriers. Thus, for any given dual variableν, there exists a solution to the inner problem that applies the same strategy on both carriers, i.e., it uses the same powers on both carriers. Since the primal recovery (3.23) combines such inner solutions to obtain an overall

7.4. Benefits of Reduced-Entropy Transmission 107

transmit strategy, we can conclude that it is optimal to perform TS between operation points that apply the same powers on both carriers. This means thatC_xˇ⁰

k, ∀kin (7.56) are scaled identity matrices, i.e.,n_k= 0, ∀k, which corresponds to proper signaling in the original system.